Singularity Theory and Algebraic Geometry
http://hdl.handle.net/20.500.11824/18
Tue, 18 Jun 2019 08:46:19 GMT2019-06-18T08:46:19ZPerverse sheaves on semi-abelian varieties -- a survey of properties and applications
http://hdl.handle.net/20.500.11824/977
Perverse sheaves on semi-abelian varieties -- a survey of properties and applications
Liu Y.; Maxim L.; Wang B.
We survey recent developments in the study of perverse sheaves on semi-abelian varieties. As concrete applications, we discuss various restrictions on the homotopy type of complex algebraic manifolds (expressed in terms of their cohomology jump loci), homological duality properties of complex algebraic manifolds, as well as new topological characterizations of semi-abelian varieties.
Wed, 01 May 2019 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9772019-05-01T00:00:00ZOn Lipschitz rigidity of complex analytic sets
http://hdl.handle.net/20.500.11824/970
On Lipschitz rigidity of complex analytic sets
Fernandes A.; Sampaio J. E.
We prove that any complex analytic set in $\mathbb{C}^n$ which is Lipschitz normally embedded at infinity and has tangent cone at infinity that is a linear subspace of $\mathbb{C}^n$ must be an affine linear subspace of $\mathbb{C}^n$ itself. No restrictions on the singular set, dimension nor codimension are required. In particular, any complex algebraic set in $\mathbb{C}^n$ which is Lipschitz regular at infinity is an affine linear subspace.
Tue, 26 Feb 2019 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9702019-02-26T00:00:00ZA jacobian module for disentanglements and applications to Mond's conjecture
http://hdl.handle.net/20.500.11824/910
A jacobian module for disentanglements and applications to Mond's conjecture
Fernández de Bobadilla J.; Nuño Ballesteros J. J.; Peñafort Sanchis G.
Let $f:(\mathbb C^n,S)\to (\mathbb C^{n+1},0)$ be a germ whose image is given by $g=0$. We define an $\mathcal O_{n+1}$-module $M(g)$ with the property that $\mathscr A_e$-$\operatorname{codim}(f)\le \dim_\mathbb C M(g)$, with equality if
$f$ is weighted homogeneous.
We also define a relative version $M_y(G)$ for unfoldings $F$, in such a way that $M_y(G)$ specialises to $M(g)$ when $G$ specialises to $g$. The main result is that if $(n,n+1)$ are
nice dimensions, then $\dim_\mathbb C M(g)\ge \mu_I(f)$, with equality if and only if $M_y(G)$ is Cohen-Macaulay, for some stable unfolding $F$. Here, $\mu_I(f)$ denotes the image
Milnor number of $f$, so that if $M_y(G)$ is Cohen-Macaulay, then Mond's conjecture holds for $f$; furthermore, if $f$ is weighted homogeneous, Mond's conjecture for $f$ is
equivalent to the fact that $M_y(G)$ is Cohen-Macaulay. Finally, we observe that to prove Mond's conjecture, it is enough to
prove it in a suitable family of examples.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9102019-01-01T00:00:00ZOn Zariski’s multiplicity problem at infinity
http://hdl.handle.net/20.500.11824/905
On Zariski’s multiplicity problem at infinity
Sampaio J. E.
We address a metric version of Zariski's multiplicity conjecture at infinity that says that two complex algebraic affine sets which are bi-Lipschitz homeomorphic at infinity must have the same degree. More specifically, we prove that the degree is a bi-Lipschitz invariant at infinity when the bi-Lipschitz homeomorphism has Lipschitz constants close to 1. In particular, we have that a family of complex algebraic sets bi-Lipschitz equisingular at infinity has constant degree. Moreover, we prove that if two polynomials are weakly rugose equivalent at infinity, then they have the same degree. In particular, we obtain that if two polynomials are rugose equivalent at infinity or bi-Lipschitz contact equivalent at infinity or bi-Lipschitz right-left equivalent at infinity, then they have the same degree.
Tue, 14 Aug 2018 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9052018-08-14T00:00:00Z